# A triangle has corners at points A, B, and C. Side AB has a length of #12 #. The distance between the intersection of point A's angle bisector with side BC and point B is #6 #. If side AC has a length of #9 #, what is the length of side BC?

Angle bisector theorem :

Given

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To find the length of side BC, we can use the Angle Bisector Theorem. According to this theorem, the length of the segment formed by the angle bisector of a triangle divides the opposite side into two segments that are proportional to the other two sides of the triangle.

Let's denote the length of side BC as x. Using the Angle Bisector Theorem, we can set up the following proportion:

[\frac{AC}{AB} = \frac{BC}{BD}]

Given that AC is 9, AB is 12, and BD is 6 (the distance between the intersection of point A's angle bisector with side BC and point B), we can plug these values into the equation:

[\frac{9}{12} = \frac{x}{6}]

Solving for x:

[x = \frac{9 \times 6}{12}]

[x = \frac{54}{12}]

[x = 4.5]

So, the length of side BC is 4.5.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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- In the figure given identify the congruent and/or similar triangles and find the value of x and y?
- A triangle has corners at points A, B, and C. Side AB has a length of #7 #. The distance between the intersection of point A's angle bisector with side BC and point B is #4 #. If side AC has a length of #14 #, what is the length of side BC?
- A triangle has corners at points A, B, and C. Side AB has a length of #38 #. The distance between the intersection of point A's angle bisector with side BC and point B is #8 #. If side AC has a length of #46 #, what is the length of side BC?
- Triangle A has sides of lengths #48 ,36 #, and #54 #. Triangle B is similar to triangle A and has a side of length #5 #. What are the possible lengths of the other two sides of triangle B?

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